Dynamical Systems Projects

Book: Nonlinear Dynamics and Chaos, by Steven Strogatz

Prerequisites: Linear algebra (e.g. MAT 211). Exposure to differential equations in some sense.

Biological systems are, more often than not, complex, interconnected, and non-linear. In addition, limited experimental data make the complete understanding of a given system difficult-to-impossible. With this in mind, many scientists turn to modeling of these systems as a way to understand them more fully. A prominent class of these models fall under dynamical systems, taking the form of a differential equation.

The purpose of this project will be to explore the field of continuous-time dynamical systems, using biological models as a guide throughout the process. We will look at phase-plane analysis, bifurcations, and introduce the concept of chaos. Supplementary topics that may be discussed, depending on the interests of the student (common modeling assumptions, integration techniques, interpretations of system characteristics in an intuitive way, etc).

Book: A Modern Introduction to Dynamical Systems, by Richard Brown

Prerequisites: A course in real analysis (e.g. MAT 319/320)

Continuous maps of the circle to itself are relatively simple to define and visualize. However, many systems of mathematical or physical interest are modeled extremely well by iterated application of such a map on a point on the circle. For example, asking how an initial point moves under the rotation map by some angle θ will lead us to an investigation of the continued fraction expansion, which is of great interest in number theory. Expanding on this will lead us to define the rotation number of a map and then to the stability of the solar system! On the other hand, the doubling map, which sends a point on the circle to a point with twice its angle from a reference point, is a fantastic example of a chaotic map. Investigation into this map leads to symbolic dynamics, connecting circle maps to linear algebra and information theory. Various connections with other parts of dynamical systems may be discussed, according to interest.

Book: The Poincare Half Plane: A Gateway to Modern Geometry, by Saul Stahl

Prerequisites: Calculus and proofs. Courses in linear algebra, topology, and complex analysis (e.g. MAT 310, 322, 342, 360, 362, 364) will allow the mentee to delve further into the material, but are not necessary.

In Euclidean geometry, one way to state the parallel postulate is to assume that given a line l and a point x not on that line, there exists exactly one line l’ containing x which does not intersect l.

For many years in the 18th and 19th centuries, mathematicians tried to deduce the parallel postulate from the other axioms of geometry, but a valid proof alluded them. Indeed, there are non-euclidean geometries (a fancy way of describing how to define the main objects in geometry like lines) which satisfy all the axioms of Euclidean geometry except the parallel postulate.

One such model is the hyperbolic ball or hyperbolic half plane. In hyperbolic space, the fastest way to get between two points may not be in a Euclidean straight line, but rather in certain circular arcs. The sum of the angles of a hyperbolic triangle is not necessarily equal to 180 degrees. In this project, we will study the basics of hyperbolic geometry in the planar setting, comparing theorems in Euclidean geometry and their hyperbolic counterparts. After getting through some basics, the direction will be up to the interests of the student.

Book: Fourier Analysis, by T. W. Korner

Prerequisites: A course in real analysis (e.g. MAT 319/320)

In the mid-1800s Joseph Fourier found that periodic functions can be realized as infinite sums of sines and cosines. This realization has made a profound impact on virtually every subfield of science and engineering. The mentee will begin by examining the sense in which these infinite sums converge, through the Dirichlet and Fejer kernels. After this, the student will delve into a number of applications of this theory, including applications to probability, Brownian motion, and dynamical systems.

Book: The Geometry of Fractal Sets, by Kenneth Falconer.

Prerequisites: A course in analysis (e.g. MAT 319/320). Enrollment in a class in measure theory (e.g. MAT 324) would be helpful, but is not necessary, as the minimal measure theory needed can be developed during the project.

We are used to the idea that points are zero dimensional, lines have one dimension, and planes have two dimensions. However, defining dimension of general sets in a careful way is not so easily done. Take for example, the Middle Thirds Cantor set, obtained by taking the unit interval and removing the middle third, and continuing this process indefinitely to the remaining intervals. What is the dimension of this set?

Such “fractal” sets occur naturally in analysis, probability, and dynamical systems, and mathematicians have developed several notions of dimension to study these sets. In this project, we will define various notions of dimension for sets in the plane, discuss some general properties of these dimensions, and use them to study interesting examples of fractal sets like the one described above. Possible directions for this project include constructing continuous yet nowhere differentiable functions, studying when fractal sets are a subset of a path in the plane, studying projections of fractal sets onto lines, or whatever other possible topic the student is interested in exploring. There are many possible directions.

Book: Complex Analysis by Theodore Gamelin

Prerequisites: A course in real analysis (e.g. MAT 319/320) and a course in complex analysis (e.g. MAT 342).

Let f_c(z)=z^2+c be a polynomial, with c ∈ C. When studying the dynamics of polynomials, one is interested in the behavior of a point under successful iterations by f. The Mandelbrot set M is defined as the set of c so that the sequence of iterates,{0,c,c^2+c,(c^2+c)^2+c,...}, is bounded. Despite this simple definition, much is still unknown about the Mandelbrot set, and therefore much is still unknown about the iteration of even these simple quadratic polynomials! In this project, we will learn how to study the dynamics of polynomials, with an emphasis on specific examples, trying to avoid any difficult general theory where we can. We will see that we can partition the plane into a stable set called the Fatou set, and a fractal chaotic set called the Julia set. We will see how the dynamics of specific polynomials f_c relates to the Mandelbrot set.

After learning the basics, we can proceed in many directions. One direction is learning how to draw computer images of the Mandelbrot set, and of Julia sets of polynomials. We would also learn the mathematics of why our programs work.

These sets have an interesting fractal structure. Students could also study the theory behind Newton’s method, a technique learned in calculus courses to find the zeros of polynomials. Students could also learn about iterating rational or general entire functions, or learn the proofs and the complex analysis behind some of the several important tools we use in this area.

Book: Dynamic Programming and Optimal Control, Vol. II, 4th Edition: Approximate Dynamic Programming, by Dimitri P. Bertsekas

Prerequisites: A course in linear algebra (e.g. MAT 211). Exposure to real analysis (e.g. MAT 319) is a bonus.

How should rational agents plan for the future? This is the underlying question asked by the Federal Reserve, your neighbor’s Roomba, the electric utility company, and the AlphaZero chess player. Each of these problems features agents interacting with uncertain, evolving environments. Agents are tasked with finding policies that optimize their outcomes with respect to particular performance criteria. In this project, the mentee will explore the theory of Markov Decision Processes, the mathematical model at the heart of this class of optimization problems. They will prove the existence of optimal policies for finite and infinite horizon MDPs under the expected total discounted reward criterion, and study algorithms which directly compute these policies. Depending on the mentee’s interests and maturity, this project can either adopt a more theoretical flavor with analysis of additional performance criteria, or a more applied flavor with reinforcement learning.